Spatial autocorrelation statistics are important and popular tools for describing spatial distributions of genetic variation in populations. Moreover, they potentially can be used for studying population genetics as a space-time process, rather than a purely temporal process. Most of the important population genetic factors, including natural selection, genetic drift, gene dispersal, and mating system, interact with spatial structure. The proposed research aims to fill in some important gaps in our knowledge of how spatial structure is changed by various combinations of these factors, and how these changes are reflected in spatial autocorrelation statistics. A major goal is to characterize the process specific and parametric variation in these statistics. Another is to characterize the stochastic and statistical sources of variation. Such an account of the relative amounts of variation in spatial autocorrelation statistics will add to our understanding of the conditions under which autocorrelation statistics can be used to detect differences in spatial structure caused by different population genetic processes, such as natural selection, that are acting on a genetic locus. I will study these properties for both autocorrelation statistics for gene frequency data, using Moran's I-statistics and the Mantel statistic, and spatial autocorrelation statistics known as join-count statistics, that are appropriate for spatial distributions of individual genotypes. One of my specific objectives is to characterize the statistical, stochastic and parametric sources of variation, for join-count statistics, I-statistics, and F-statistics for genetic loci under isolation by distance. Another is a similar characterization for loci that are under certain specific forms of natural selection. I also will explore isolation by distance for a polygenic or quantitative trait. Two types of population models will be studied, the continuous population model and the stepping stone type models. Much of the results will be obtained by contrasting the properties of the spatial statistics for sets of replicate Monte Carlo simulation runs, where there are different processes and parameter values for each set. Where possible these results will be compared to existing mathematical models of population genetics. In addition, some new mathematical models will be developed, and analyzed using new numerical methods.